Multiphase averaging for generalized flows on manifolds

We present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in R^m and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.